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Theorem lejoin1 18381
Description: A join's first argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.)
Hypotheses
Ref Expression
joinval2.b 𝐡 = (Baseβ€˜πΎ)
joinval2.l ≀ = (leβ€˜πΎ)
joinval2.j ∨ = (joinβ€˜πΎ)
joinval2.k (πœ‘ β†’ 𝐾 ∈ 𝑉)
joinval2.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
joinval2.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
joinlem.e (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ )
Assertion
Ref Expression
lejoin1 (πœ‘ β†’ 𝑋 ≀ (𝑋 ∨ π‘Œ))

Proof of Theorem lejoin1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 joinval2.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 joinval2.l . . 3 ≀ = (leβ€˜πΎ)
3 joinval2.j . . 3 ∨ = (joinβ€˜πΎ)
4 joinval2.k . . 3 (πœ‘ β†’ 𝐾 ∈ 𝑉)
5 joinval2.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
6 joinval2.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐡)
7 joinlem.e . . 3 (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ )
81, 2, 3, 4, 5, 6, 7joinlem 18380 . 2 (πœ‘ β†’ ((𝑋 ≀ (𝑋 ∨ π‘Œ) ∧ π‘Œ ≀ (𝑋 ∨ π‘Œ)) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧) β†’ (𝑋 ∨ π‘Œ) ≀ 𝑧)))
98simplld 766 1 (πœ‘ β†’ 𝑋 ≀ (𝑋 ∨ π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3057  βŸ¨cop 4636   class class class wbr 5150  dom cdm 5680  β€˜cfv 6551  (class class class)co 7424  Basecbs 17185  lecple 17245  joincjn 18308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-lub 18343  df-join 18345
This theorem is referenced by:  joinle  18383  latlej1  18445
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